Exactly computing bivariate projection depth contours and median

TL;DR

This paper introduces a simplified method for exactly computing bivariate projection depth contours and median, enhancing practical application of robust multivariate data analysis.

Contribution

It derives a simple form of the projection depth function for (Med, MAD), enabling exact computation of depth contours and median in bivariate data.

Findings

Derived a simple form of the projection depth function.

Extended exact computation from points to contours and median.

Facilitated practical application of projection depth methods.

Abstract

Among their competitors, projection depth and its induced estimators are very favorable because they can enjoy very high breakdown point robustness without having to pay the price of low efficiency, meanwhile providing a promising center-outward ordering of multi-dimensional data. However, their further applications have been severely hindered due to their computational challenge in practice. In this paper, we derive a simple form of the projection depth function, when (\mu, \sigma) = (Med, MAD). This simple form enables us to extend the existing result of point-wise exact computation of projection depth (PD) of Zuo and Lai (2011) to depth contours and median for bivariate data.